Recent results from LEP on mixing in the $B$ sector are reviewed. The new LEP $95\%$ CL lower limit on $B_s$ mixing of $\Delta m_s > 9.5~{\rm ps^{-1}}$ combined with the LEP average for $\Delta m_d$ of $0.469 \pm 0.017~{\rm ps^{-1}}$, starts to significantly constrain the unitarity triangle for the first time

Coyle, P

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B MIXING AT LEPP. COYLECERN, Geneva, Switzerland andCentre de Physique des Particules De Marseille, FranceRecent results from LEP on mixing in the B sector are reviewed. The new LEP95% CL lower limit on Bsmixing of ms> 9:5 ps 1combined with the LEPaverage for mdof 0:4690:017ps 1, starts to signicantly constrain the unitaritytriangle for the rst time.1 IntroductionThe observed avour eigenstates of the neutral Bqmesons (Bdor Bs) are linearcombinations of their corresponding mass eigenstates. The mass dierence(mq) between the mass eigenstates gives rise to a time dependent phasedierence between their wavefunctions and thus a probability to observe a B0orB0which oscillates as a function of time. Assuming CP is conserved andnegligible dierence in decay widths between the two mass eigenstates, theprobability of observing a Bqmeson in a state the same (P+, unmixed) ordierent (P , mixed) from that at production is given byP(t) =12Bqe t=Bq(1 cos(mqt))where t is the proper time and Bqis the relevant Bqlifetime. A measurementof the oscillation frequency therefore allows mqto be determined, in analogyto the neutral kaon system.Within the Standard Model, mixing is mediated by box diagrams involvingtwo W -bosons and two quarks of charge23e. Taking only the dominant topquark exchange into account, the mass dierence is given bymq=G2F62mBqm2tFm2tm2WBBBqf2BqjVtbVtqj2(1)where GFis the Fermi constant, F gives the functional dependence on the ratioof the top quark and W masses (mtand mW), Bis a known QCD correctionfactor (0.55), mBqis the mass of the Bqmeson, BBqis the `bag' factor andfBqthe decay constant for the Bqmeson. Substantial theoretical uncertaintiesin the parameters hinder the extraction of the product of the CKM matrixelements jVtbVtqj from a measurement of the oscillation frequency.1γραβVcbtdVλ=VVts1~Vλ V*(ρ,η)η(1,0)(0,0)tdubcbm d∆sm∆λ λFigure 1: The unitarity triangle represented in     space;  and  are parameters of theWolfenstein parameterization of the CKM matrix and  is the sin of the Cabbibo angle(0.2205). The side opposite the angle  is constrained by ms=md.The ratiomsmd=mBsmBd2VtsVtd2(2)is under better control as all dependence on the top mass drops out and theratio of the theoretical quantities  =pBBsfBspBBdfBdis better known (1:150:051).The ratio ms=mdis of particular interest, as assuming jVcbj = jVtsj, it iscan be used to constrain the length of a side of one of the most useful unitaritytriangles of the CKM matrix (Fig. 1). Assuming the Standard Model to becorrect, existing measurements of the CKM matrix elements can be used1topredict a preferred range for msof 8:6 ps 1< ms< 17:0 ps 1.2 Experimental MethodsAt LEP, both time-integrated and time-dependent methods can be used to ex-tract information on mdand ms. As time-integrated measurements haveno sensitivity to large m, for the fast Bsoscillation one must rely on mea-suring the time dependence of the mixing. At the (4S) only the total timeintegrated probability of Bdmixing can be measured as the B's are producedat rest and Bsmesons are not created.Experimental measurement of the oscillation frequency necessitate taggingthe production and decay states of the Bqmeson and a measurement of itsproper time.2The decay tag is obtained by looking at the charge of some of the nal statedecay particles for example, a high ptlepton or the charge of a reconstructedcharm particle (Dfor Bdmixing or Dsfor Bsmixing).The production tag is more involved and is inferred from studying theproperties of the b hadron in the opposite thrust hemisphere to the decaytag for example, the charge of a high ptlepton or the momentum weightedhemisphere jet charge. Many analyses also use information from the samehemisphere, such as the charge of the leading fragmentation kaon (just for Bs)or the same hemisphere jet charge. Although the rate of mistag () from acombination of the same and opposite jet charges (  28%) is usually higherthan that from a high ptlepton (  20%) or fragmentation kaon (  26%)this is compensated by its high eciency. Using a method (\optimal tagging")of combining information from variables choosen to have some discriminatingpower to identify the initial state, (for example the ptof a lepton, the value ofthe jet charge etc.) ALEPH2has obtained an eective (event-by-event) mistagof  27% with 100% eciency.The proper time is calculated as t = g` where g is 1/boost and ` is thedecay length. The decay length is the distance from the primary vertex tothe B decay vertex projected onto the estimated B ight direction. The Bvertex is reconstructed by combining the charged track(s) thought to comedirectly from the B with a fully or partially reconstructed charm particle. Theresolution on the proper time can be expressed ast=q(tg=g)2+ (g`)2:The rst term corresponds to the eect of the boost resolution (g=g  10  20%) and increases with t. The second term, approximately constant, is due tothe decay length resolution (` 230  400m) and dominates the resolutionat small t.The nal sensitivity of a particular analysis to mixing depends on the\properties" of the analysis in the following way3signal=noise rN2(1  2)p exp ( 12(mqt)2)where N is the total number of selected events and p is the purity of thesample. As the exponential term, due to the smearing from the proper timeresolution, is important for large mq, the requirements on the proper timeresolution for Bsmixing are much more stringent than for Bdmixing. For themore exclusive analyses the decrease in statistics is usually compensated byimprovements in purity, mistag and proper time resolution.33 mdResultsFigure 2 summarises the latest LEP measurements of md. Also show forcomparison are measurements from the (4S) (using Bd= 1:570:05 ps) andCDF. The names of the analyses are chosen such that the words before (after)the slash refer to the method for tagging the decay (production) state. Here Lmeans high ptlepton, QJ means jet charge, OPT means optimal tagging, Dmeans fully reconstructed Dmeson and nally L means a new decay tagfrom DELPHI in which a slow pion from the Dis vertexed with a lepton. Forthe LEP average, only measurements plotted with a lled circle are used, alsono attempt is made to take into account the small statistical and systematiccorrelations. Figure 3 shows a selection of ts for the mdmeasurements.4 msResultsNone of the Bsmixing analyses claim to measure ms; instead a 95% con-dence level exclusion is quoted. The exclusion is determined by nding the min-imum in the negative log likelihood distribution as a function of the assumedvalue of msand moving up from this minimum by a xed number of likeli-hood units (lnL95). Naively, for a 95% CL one would expect lnL95= 1:92;in practice it is found that, due to the presence of fake minima, the value islarger ( 2-3), may be a function of the assumed msand dier from analysisto analysis. A calibration of the 95% CL curve must therefore be performed,at each value of ms, by studying many fast Monte Carlo experiments.To estimate the systematic uncertainties, when generating each fast MonteCarlo experiment used to calculate the 95% CL curve, ALEPH simultaneouslyvary the central values of all the parameters under study within their Gaussianerror. The net eect is that the lnL95is slightly increased ( 0:3   0:5).OPAL have a dierent approach; they introduce a Gaussian constraint oneach of the systematic parameters and maximise the likelihood with respect tothese constrained parameters at each value of ms. In general the systematicsare small and mostly aect the low msregion. For the inclusive analyses themost important systematic uncertainty is from fs, the fraction of Bseventsproduced when the b quarks from the Z0hadronise. For the Dsbased analysisthe systematic from fsB(Bs! DsX) dominates.Figure 4 summarises the 95% CL exclusion regions obtained from all theLEP analyses using this procedure. As the quoted exclusion regions are inu-enced by statistical uctuations, the msreach of the dierent analyses cannot be easily compared.The ALEPH DsL/OPT analysis2based on only 277 Ds-lepton candidatesreconstructed in 6 channels (;K0K;K0K; ` ;K0K ;  + ),4Figure 2: Summary of the LEP mdmeasurements. The number in square brackets refersto the relevant reference. The data sample used is also indicated.proper time  [ps]L3allb→lb→c→lb→fake udsc( N++  + N-- ) / Ntotal00.10.20.30.40.50.6-2 0 2 4 6 8 10 12 1400.20.40.60.811.2-2 0 2 4 6 8 10t (ps)ROPAL0.10.20.30.40.50.60.70.8-2 0 2 4 6 8 1000.010.020.030.040.050.060.07-2 -1 0 1 2 3 4 5 6 7 8Measured Time (ps)Lepton-signed Jet Charge  QlHDataResult of the fitALEPHFigure 3: Likelihood ts to the fraction of events tagged as mixed versus proper time forsome mdanalyses: the L3 L/L, the OPAL D/L, the DELPHI L/QJ and the ALEPHL/QJ.5yields a limit of 6:6 ps 1. The small statistics is compensated by the very highBspurity ( 65%), and the good vertex (` 230 m) and momentumresolutions (g=g  10%).ALEPH have developed another new analysis11(Fig. 5) based on vertexingthe Dswith an oppositely charged hadronic track selected to be likely tocome from the Bsvertex (DsH/OPT). Compared to the previous analysis,this approach suers from additional backgrounds due to Dsmesons fromdirect charm production and the misassignment of tracks from the primaryvertex to the B vertex.The ALEPH L/L analysis12uses 9710 leptons with a proper time mea-surement and assuming fs= (10:2 1:6)% excludes ms< 5:6 ps 1. In theALEPH L/QJ analysis13, after events used in the L/L analysis are removedto ensure statistical independence, a total of 63131 leptons are tagged using ajet charge based on information from both hemispheres.All the DELPHI analyses14use data up to 1994. As they have not cali-brated their 95% CL a conservative value of lnL95= 3 is used here to denetheir exclusion regions. They assume fs= (10:0  2:2)%. In the DELPHIDsL/QJ analysis, the channel Ds! f0(980)+is reconstructed in additionto those used by ALEPH. The DELPHI L/L analysis uses 4778 events andprovides a limit of 2:2 ps 1. The DELPHI L/QJ analysis uses 60381 taggedevents and provides a limit of 3:6 ps 1.OPAL have calibrated their 95% CL curves and assume fs= (12 3:6)%.The OPAL DsL/QJ analysis16, which also includes a sample requiring only a from the charm particle, has a log likelihood distribution with a large dipnear 0:5 ps 1. The OPAL L/QJ analysis7uses over 90000 events and providesa limit of 3:1 ps 1. Their L/L analysis8use data only up to 1993 and gives alimit of 2:2 ps 1.4.1 Combination of msResultsIn order to combine the various analyses, naively one would simply add theindividual log likelihoods, nd the new minimum and apply the same prescrip-tion as previously discussed. In this case the calibration of the 95% CL limitcurve is rather dicult, as strictly it would require a \grand" Monte Carlosimulation incorporating all the individual analyses. This becomes even moreproblematic when attempting to combine likelihoods from dierent experi-ments. A new procedure to combine mixing results (\amplitude method"), inwhich the amplitude of the oscillation is tted as a function of ms, has beenproposed to overcome this problem3. As many of the analyses do not providethe necessary information to perform an amplitude method combination, for6Figure 4: A summary of the 95% CL exclusion regions for ms. The number in squarebrackets refers to the relevant reference. The data sample used is also indicated.00.10.20.30.40.50.60.70.80.91-2 -1 0 1 2 3 4 5 6 7 8proper time [ps]tagged-as-mixed fraction ALEPH0123456789100 2 4 6 8 10 12ALEPHMC average (∆ms=∞)data ∆ms[ps-1] ∆log(L)95% CL curve (no syst.)95% CL curve (with syst.)Figure 5: The ALEPH DsH/OPT analysis: (a) Fraction of candidates tagged as mixed as afunction of the proper time. The solid curve is the likelihood t for ms. (b) The negativelog-likelihood with respect to the minimum as a function of ms. The solid curve showsthe data. The dotted (dot-dashed) line is the 95% CL curve with (without) systematics.7Figure 6: The individual log likelihoods distributions and their sum for the various analyses.(a) ALEPH analyses, (b) DELPHI analyses, (c) OPAL analyses, (d) LEP combined.this paper, the naive \addition of log likelihood" method is adopted assuminga conservative lnL95= 3.The individual log likelihoods and their sum, relative to the minimum,for the ALEPH, DELPHI and OPAL analyses are shown in Figs. 6(a),(b)and (c) respectively. The combined ALEPH/DELPHI/OPAL lower limits arems> 7:8=6:5=4:6 ps 1. Possible statistical overlap in the DELPHI L/L andL/QJ analysis is ignored. Fig. 6(d) shows the sum of the ALEPH, DELPHI andOPAL combined log likelihoods. Again assuming lnL95= 3 the combinedLEP average is > 9:5 ps 1. Using the world average Bs= 1:61 0:10 ps17this corresponds to xs> 15:3.5 ConclusionThe measurement of mdcontinues to improve; the latest LEP average ismd= 0:4690:017 ps 1, i.e. a relative error of 3:6%. This can be comparedwith the relative theoretical error on the quantity f2BdBBd, used in equation 1to access jVtdj, of  40%1. In order to reduce the theoretical uncertainties, theemphasis of the mixing analyses is to measure the ratio ms=md. The latestLEP limit on ms> 9:5 ps 1gives ms=md> 19:5 which, via equation 2,implies jVts=Vtdj > 3:8. In terms of the unitarity triangle this corresponds to8Figure 7: The constraint in the     plane from the LEP bound ms=md> 19:5. Thethree curves correspond to  = 1:2 (solid),  = 1:15 (dashed) and  = 1:1 (dots). The regionto the left of the curves are excluded. The allowed region for the apex is taken from Ref. [1].a circle in the     plane which starts to signicantly constrain the allowedregion for its apex (Fig. 7).The Bsmixing analyses at LEP are statistics limited. As LEP has re-cently increased its energy to the W+W threshold, where Bsmesons are nolonger copiously produced, future prospects for signicantly improved limits orpossible observation of Bsmixing at LEP are not bright. Nevertheless, somemodest progress can be expected as analyses improve and all experiments in-corporate the 1995 data (+20%). The conclusive observation of the elusiveBsoscillation may therefore have to wait for future experiments at Tevatron,HERA and LHC.AcknowledgementsI am grateful to the LEP contact persons for providing their results. It is apleasure to thank Olivier Schneider for useful discussions.References1. A. Ali and D. London, \CP Violation and Flavour Mixing in the StandardModel-1996 Update", DESY 96-140, presented at the QCD Euroconfer-ence 96, Montpellier.92. D. Buskulic et al, (ALEPH Collab.), \Study of the B0sB0soscillationfrequency using D s`+combinations in Z decays", CERN-PPE/96-30,to appear in Phys. Lett. B.3. H.-G.Moser, A. Roussarie, \Mathematical Methods for B0B0OscillationAnalyses", submitted to Nucl. Inst. Meth. A.4. D. Buskulic et al, (ALEPH Collab.), \Improved measurement of the B0d B0doscillation frequency", CERN-PPE/96-102, submitted to Z. Phys. C.5. P. Checchia et al, (DELPHI Collab.), \Measurement of B0d B0doscil-lations", contribution PA1-038 to the 28th International Conference onHigh Energy Physics, Warsaw, Poland, 25-31 July 1996.6. G. Alexander et al, (OPAL Collab.), \A Measurement of the B0dOscil-lation Frequency using Leptons and DMesons", CERN-PPE/96-074,submitted to Z. Phys. C.7. OPAL Collab., \A Study of B Meson Oscillations using Inclusive LeptonEvents", contribution PA08-011 to the 28th International Conference onHigh Energy Physics, Warsaw, Poland, 25-31 July 1996.8. R. Akers et al, (OPAL Collab.), Z. Phys. C66 (1995) 543.9. M. Acciarri et al. (L3 Collab.), \Measurement of the B0dMeson Oscilla-tion Frequency", CERN-PPE/96-73, submitted to Phys. Lett. B.10. E. Meschi (CDF Collab.), \B physics at CDF", FERMILAB-CONF-96/013-E, proceedings of the conference on Production and Decay ofHyperons, Charm and Beauty Hadrons, Strasbourg, Sept. 1995.11. ALEPH Collab., \Search for Bsoscillations using inclusive Dsevents",contribution PA05-054 to the 28th International Conference on High En-ergy Physics, Warsaw, Poland, 25-31 July 1996.12. ALEPH Collab., \Measurement of the B0d B0doscillation frequency",contribution EPS0409 to the International Europhysics Conference onHigh Energy Physics, Brussels, Belgium, July 27-Aug. 2 1995.13. S. Emery, private communication.14. A. Borgland et al, (DELPHI Collab.), \Search for B0s B0soscillations",contribution PA1-039 to the 28th International Conference on High En-ergy Physics, Warsaw, Poland, 25-31 July 1996.15. OPAL Collab., \OPAL combined study of BsOscillations", contributionPA08-010 to the 28th International Conference on High Energy Physics,Warsaw, Poland, 25-31 July 1996.16. OPAL Collab., \Investigation of the BsOscillation Frequency using `and Ds` Correlations", contribution PA08-014 to the 28th InternationalConference on High Energy Physics, Warsaw, Poland, 25-31 July 1996.17. R. M. Barret et al. (Particle Data Group), Phys. Rev. D 54 (1996) 1,preview available on WWW (http://pdg.lbl.gov).10

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